I have a question about the recipe "Griddled Polenta Cakes with Caramelized Onions, Goat Cheese, and Honey" from arielleclementine. Why are my cakes falling apart while cooking them. No matter how long I left them they just turned to mush?

After reading the recipe I would guess that the polenta is undercooked and/or under-stirred.
The creaminess of warm polenta and the firmness of cooled polenta come from the corn's natural starches. When the starches are cooked in water, gelatanized, they dissolve and thicken the liquid. When the gelatinided starches cool, they form a solid matrix and hold the polenta together. The reason for stirring the polenta is to agitate the relatively large corn meal granules so they release and evenly distribute the starch into the liquid (and to avoid burning). If the mixture is not thoroughly mixed and cooked long enough, then not all of the available starch will be released and gelatanized. You may need to cook and stir longer than the five minutes called for in the recipe. The mixture should look creamy and feel smooth before you pour it out. In fact you can test it as you go by rubbing some of the soft polenta between your fingers (take a small amount out and let it cool a bit unless you have asbestos fingers). If you feel tiny hard grains then keep cooking. If it is silky smooth paste with soft, rubbery grains then you are ready to cool it. Let it cool completely in the mold and then it should stay nice and firm.